The oil and gas products that are contained, for example, in sandstone earth formations, occupy pore spaces in the rock. The pore spaces are interconnected and have a certain permeability, which is a measure of the ability of the rock to transmit fluid flow. When some damage has been done to the formation material immediately surrounding the bore hole during the drilling process or if permeability is low, a hydraulic fracturing operation can be performed to increase the production from the well. Hydraulic fracturing is a process by which a fluid under high pressure is injected into the formation to split the rock and create fractures that penetrate deeply into the formation. These fractures create flow channels to improve the near term productivity of the well.
Evaluating physical parameters of a reservoir play a key part in the appraisal of the quality of the reservoir. However, the delays linked with these types of measurements are often very long and thus incompatible with the reactivity required for the success of such appraisal developments.
One of the reasons is the complexity of a multilayer environment, it increases as the number of layers with different properties increases. Layers with different pore pressure, fracture pressure, and permeability can coexist in the same group of layers. The biggest detriment for investigating layer properties is a lack of cost-effective diagnostics for determining layer permeability, and fracture-face resistance of reservoir.
Numerous analyses have been carried out to evaluate physical parameters of a reservoir. More particularly, before-closure pressure-transient analysis has been commonly used to estimate permeability and fracture-face resistance from the pressure decline following a fracture-injection/falloff test in the reservoir.
Before-closure pressure-transient analysis is described by Mayerhofer and Economides in a paper SPE 26039 “Permeability Estimation From Fracture Calibration Treatments,” presented at the 1993 Western Regional Meeting, Anchorage, Ak., 26–28 May 1993; also by Mayerhofer, Ehlig-Economides, and Economides in a journal JPT (March 1995) on page 229 “Pressure-Transient Analysis of Fracture-Calibration Tests”; and by Ehlig-Economides, Fan, and Economides in a paper SPE 28690 “Interpretation Model for Fracture Calibration Tests in Naturally Fractured Reservoirs” presented at the 1994 SPE International Petroleum Conference and Exhibition of Mexico, 10–13 Oct. 1994. The analysis was formulated in part using the early-time infinite-conductivity fracture solution of the partial differential equation that Gringarten, Ramey, and Raghavan suggested in a journal SPEJ (August 1974) on page 347 “Unsteady-State Pressure Distributions Created by a Well With a Single Infinite-Conductivity Vertical Fracture” which assumed the use of a slightly compressible reservoir fluid. However, diagnostic fracture-injection/falloff tests are commonly implemented in reservoirs containing highly compressible fluids, for example, in natural gas reservoirs. When the compressibility of the reservoir fluid deviates from the assumption of a slightly compressible fluid, the analysis methods as used in the prior art can lead to erroneous permeability and fracture-face resistance estimates.
The errors in the estimates of the permeability and fracture-face resistance are significant and can be detected in the plotting of the experimental data obtained with a slightly compressible reservoir fluid. As a matter of fact, these errors are the consequences of the inaccuracy of the approximations as used in the prior art. These approximations used in connection with the actual theory developed with the pressure-transient leakoff analysis are based on the assumption that the reservoir fluid properties are not functions of pressure, which could not be the case when the reservoir fluid is a gas. The approximations as assumed in the prior art are as follows:
1) Before-Closure Pressure-Transient Leakoff Analysis Assuming a Slightly-Compressible Reservoir Fluid
The pressure decline following a fracture-injection/falloff test can be divided into two distinct regions: before-fracture closure and after-fracture closure. Before-closure pressure-transient analysis is used to determine permeability from the before-fracture closure decline data. Mayerhofer and Economides in paper SPE 26039 divide the before-closure pressure difference between a point in an open, infinite-conductivity fracture and a point in the undisturbed reservoir into four components written as:Δp(t)=Δpres(t)+Δpcake(t)+Δppiz(t)+Δpfiz(t).  (1)
The pressure difference in the polymer invaded zone, Δppiz(t), the filtrate invaded zone, Δpfiz(t), and across the filtercake, Δpcake(t), can be grouped into a fracture-face pressure difference term, Δpface(t). Consequently, the pressure gradient consists of reservoir and fracture-face resistance components, and is written as:Δp(t)=Δpres(t)+Δpface(t).  (2)2) Fracture-Face Pressure Difference
In the same way, in paper SPE 26039 Mayerhofer and Economides determine the fracture-face resistance pressure difference by using the concept of a fracture-face skin proposed by Cinco-Ley and Samaniego in paper SPE 10179 “Transient Pressure Analysis: Finite Conductivity Fracture Case Versus Damage Fracture Case” presented at the 1981 SPE Annual Technical Conference and Exhibition, San Antonio, Tex., 5–7 Oct. 1981. Cinco-Ley and Samaniego defined fracture-face skin as:
                                          s            f                    =                                                    π                ⁢                                                                  ⁢                                  b                  fs                                                            2                ⁢                                  L                  f                                                      ⁡                          [                                                k                                      k                    fs                                                  -                1                            ]                                      ,                            (        3        )            where bfs is the damaged zone width, Lf is the fracture half-length, k is the reservoir permeability, and kfs is the damaged-zone permeability. Mayerhofer and Economides account for variable fracture-face skin by defining resistance, in paper SPE 26039, as:
                                                        R              fs                        ⁡                          (              t              )                                =                                                    b                fs                            ⁡                              (                t                )                                                    k              fs                                      ,                            (        4        )            and dimensionless resistance in journal JPT of (March 1995) by:
                                                        R              D                        ⁡                          (              t              )                                =                                                                      R                  fs                                ⁡                                  (                  t                  )                                                            R                0                ′                                      ≈                                          t                                  t                  ne                                                                    ,                            (        5        )            where R′0 is the reference filtercake resistance at the end of the injection and tne is the time at the end of the injection.
With Eqs. 4 and 5, fracture-face skin is written as:
                                          s            f                    =                                                                      π                  ⁢                                                                          ⁢                                      kR                    0                    ′                                    ⁢                                                            R                      D                                        ⁡                                          (                      t                      )                                                                                        2                  ⁢                                      L                    f                                                              -                                                π                  ⁢                                                                          ⁢                                      b                    fs                                                                    2                  ⁢                                      L                    f                                                                        ≅                                          π                ⁢                                                                  ⁢                                  kR                  0                  ′                                ⁢                                                      R                    D                                    ⁡                                      (                    t                    )                                                                              2                ⁢                                  L                  f                                                                    ,                            (        6        )            or as:
                              s          f                =                                            π              ⁢                                                          ⁢                              kR                0                ′                                                    2              ⁢                              L                f                                              ⁢                                                    t                                  t                  ne                                                      .                                              (        7        )            
Fracture-face skin is equivalent to a dimensionless pressure difference across the fracture face; thus, it can be written as:
                                          p                                          L                f                            ⁢              D                                =                                                                      kh                  p                                ⁢                Δ                ⁢                                                                  ⁢                                  p                  face                                                            141.2                ⁢                                  q                                      L                    f                                                  ⁢                B                ⁢                                                                  ⁢                μ                                      =                          s              f                                      ,                            (        8        )            where hp is the permeable reservoir thickness, qLf is the total injection (leakoff) rate into both wings of the hydraulic fracture, B is the formation volume factor of the filtrate, and μ is the filtrate viscosity. With Eq. 8, the fracture-face pressure difference is written as:
                              Δ          ⁢                                          ⁢                      p            face                          =                  141.2          ⁢                      (            π            )                    ⁢                                    μ              ⁢                                                          ⁢                              R                0                ′                                                                    h                p                            ⁢                              L                f                                              ⁢                                    q                                                L                  f                                ⁢                B                                      2                    ⁢                                                    t                                  t                  ne                                                      .                                              (        9        )            
With a fracture symmetric about the wellbore, the total injection (leakoff) rate can be written as:qLfB=2ql.  (10)where ql is the leakoff rate in one wing of the fracture. The fracture-face pressure difference is written as:
                              Δ          ⁢                                          ⁢                      p            face                          =                  141.2          ⁢                      (            π            )                    ⁢                                    μ              ⁢                                                          ⁢                              R                0                ′                                                                    h                p                            ⁢                              L                f                                              ⁢                      q            ℓ                    ⁢                                                    t                                  t                  ne                                                      .                                              (        11        )            
Define:R0≡μR′0,  (12)where R0 is the fracture-face resistance, then the fracture-face pressure difference is written as:
                              Δ          ⁢                                          ⁢                      p            face                          =                  141.2          ⁢                      (            π            )                    ⁢                                    R              0                                                      h                p                            ⁢                              L                f                                              ⁢                      q            ℓ                    ⁢                                                    t                                  t                  ne                                                      .                                              (        13        )            
Assuming the fracture-face skin is a steady-state skin, the pressure difference at the fracture face at any time since the injection began is written as:
                                          (                          Δ              ⁢                                                          ⁢                              p                face                                      )                    n                =                  141.2          ⁢                      (            π            )                    ⁢                                    R              0                                                      h                p                            ⁢                              L                f                                              ⁢                                    (                              q                ℓ                            )                        n                    ⁢                                                                      t                  n                                                  t                  ne                                                      .                                              (        14        )            where the subscript n denotes a time tn.
According to Nolte, K. G. in a journal SPEFE (December 1986): “A General Analysis of Fracturing Pressure Decline With Application to Three Models,” on page 571, the leakoff rate from one wing of a hydraulic fracture during a shut-in period is written as:
                                          (                          q              ⁢                                                          ⁢              l                        )                    j                =                                            -                              [                                  24                  5.165                                ]                                      ⁢                                                                                A                    f                                                        S                    f                                                  ⁡                                  [                                                            ⅆ                                              (                                                  Δ                          ⁢                                                                                                          ⁢                          p                                                )                                                                                    ⅆ                                              (                                                  Δ                          ⁢                                                                                                          ⁢                          t                                                )                                                                              ]                                            j                                ≅                                    [                              24                5.165                            ]                        ⁢                                          A                f                                            S                f                                      ⁢                                                            (                                                            p                                              j                        -                        1                                                              -                                          p                      j                                                        )                                                  (                                                            t                      j                                        -                                          t                                              j                        -                        1                                                                              )                                            .                                                          (        15        )            where Af is the fracture area, Sf is the fracture stiffness and the subscript j is a time index. Sf can be determined using Table 1 which summarizes what Valkó and Economides determine in Chap. 2, pages 19–51: “Linear Elasticity, Fracture Shapes, and Induced Stresses,” Hydraulic Fracture Mechanics, John Wiley & Sons, New York City (1997). The fracture stiffness Sf for 2D fracture models can be calculated by using either one of the three formulas as shown in Table 1, the radial equation, the Perkins-Kern-Nordgren equation, or the Geertsma-deKlerk equation.
Define:
                                          d            j                    ≡                                    (                                                p                                      j                    -                    1                                                  -                                  p                  j                                            )                                      (                                                t                  j                                -                                  t                                      j                    -                    1                                                              )                                      ,                            (        16        )            then the leakoff rate from one wing can be written as:
                                          (                          q              ⁢                                                          ⁢              l                        )                    j                =                              24            5.165                    ⁢                                    A              f                                      S              f                                ⁢                                    d              j                        .                                              (        17        )            
At any time during the shut-in period, tn>tne, the fracture-face pressure difference is written as:
                                          (                          Δ              ⁢                                                          ⁢                              p                face                                      )                    n                =                                            141.2              ⁢                              (                π                )                            ⁢              24                        5.615                    ⁢                                    A              f                                                      h                p                            ⁢                              L                f                                              ⁢                                    R              0                                      S              f                                ⁢                      d            n                    ⁢                                                                      t                  n                                                  t                  ne                                                      .                                              (        18        )            
The ratio of permeable fracture area to total fracture area is defined by:
                                          r            p                    ≡                                    A              p                                      A              f                                      ,                            (        19        )            where for a rectangular-shaped fracture, Ap=hpLf, and the fracture-face pressure difference at any time during the shut-in period, tn>tne, is written as:
                                          (                          Δ              ⁢                                                          ⁢                              p                face                                      )                    n                =                                            141.2              ⁢                              (                π                )                            ⁢              24                        5.615                    ⁢                                    R              0                                                      r                p                            ⁢                              S                f                                              ⁢                      d            n                    ⁢                                                                      t                  n                                                  t                  ne                                                      .                                              (        20        )            
Eq. 20 is also applicable to radial, elliptical, or other idealized fracture geometry by defining fracture-face skin in terms of equivalent fracture half-length, Le, and noting that any fracture area can be expressed in terms of an “equivalent” rectangular fracture area.
3) Reservoir Pressure Difference
As in previously mentioned article of the journal SPEJ (August 1974) on page 347: “Unsteady-State Pressure Distributions Created by a Well With a Single Infinite-Conductivity Vertical Fracture”, the pressure drop in the reservoir is modeled by Gringarten, Ramey, and Raghavan for a slightly-compressible fluid, and is written in dimensionless form as:PLfD=√{square root over (πtLfD)},  (21)where
                                          p                                          L                f                            ⁢              D                                =                                    k              ⁢                                                          ⁢                              h                p                            ⁢              Δ              ⁢                                                          ⁢                              p                res                                                    141.2              ⁢                              q                                  L                  f                                            ⁢              B              ⁢                                                          ⁢              μ                                      ,                            (        22        )            and
                              t                                    L              f                        ⁢            D                          =                  0.0002637          ⁢                                                    k                ⁢                                                                  ⁢                t                                            ϕ                ⁢                                                                  ⁢                μ                ⁢                                                                  ⁢                                  c                  t                                ⁢                                  L                  f                  2                                                      .                                              (        23        )            
In Eq. 23, φ is the porosity and ct is the total compressibility. Equating Eqs. 21 and 22 and combining with Eq. 10 results in:
                              B          ⁢                                          ⁢          Δ          ⁢                                          ⁢                      p            res                          =                  141.2          ⁢                      (            2            )                    ⁢                                    B              ⁢                                                          ⁢              μ                                      k              ⁢                                                          ⁢                              h                p                                              ⁢          q          ⁢                                          ⁢          l          ⁢                                                    π                ⁢                                                                  ⁢                                  t                                                            L                      f                                        ⁢                    D                                                                        .                                              (        24        )            
By expanding the dimensionless time term, the reservoir pressure difference can be written as:
                              Δ          ⁢                                          ⁢                      p            res                          =                  141.2          ⁢                      (            2            )                    ⁢                      (            0.02878            )                    ⁢                      1                                          h                p                            ⁢                              L                f                            ⁢                              k                                              ⁢                                    μ                              ϕ                ⁢                                                                  ⁢                                  c                  t                                                              ⁢          q          ⁢                                          ⁢          l          ⁢                                          ⁢                                    t                        .                                              (        25        )            
The pressure difference at any time tn is written using superposition as:
                                          (                          Δ              ⁢                                                          ⁢                              p                res                                      )                    n                =                  141.2          ⁢                      (            2            )                    ⁢                      (            0.02878            )                    ⁢                      1                                          h                p                            ⁢                              L                f                            ⁢                              k                                              ⁢                                    μ                              ϕ                ⁢                                                                  ⁢                                  c                  t                                                              ⁢                                    ∑                              j                =                1                            n                        ⁢                                          [                                                                            (                                              q                        ℓ                                            )                                        j                                    -                                                            (                                              q                        ℓ                                            )                                                              j                      -                      1                                                                      ]                            ⁢                                                                                          t                      n                                        -                                          t                                              j                        -                        1                                                                                            .                                                                        (        26        )            
In a simplification of the more general method, Mayerhofer and Economides in paper SPE 26039, and Valkó and Economides in a journal SPEPF (May 1999) on page 117: “Fluid-Leakoff Delineation in High-Permeability Fracturing”, assume that during the injection, the first ne+1 leakoff rates are constant, where ne is the index corresponding to the time at the end of the injection and the beginning of the pressure falloff, the leakoff rates can be written as:(ql)j=Constant 1≦j≦ne+1, and (ql)0=0.  (27)
With Eq. 27, the reservoir pressure difference at any time tn is written as:
                                          (                          Δ              ⁢                                                          ⁢                              p                res                                      )                    n                =                  141.2          ⁢                      (            2            )                    ⁢                      (            0.02878            )                    ⁢                      1                                          h                p                            ⁢                              L                f                            ⁢                              k                                              ⁢                                                                      μ                                      ϕ                    ⁢                                                                                  ⁢                                          c                      t                                                                                  ⁡                              [                                                                                                                                                                                        (                                                              q                                ℓ                                                            )                                                        1                                                    ⁢                                                                                    t                              n                                                                                                      +                                                                              [                                                                                                                            (                                                                      q                                    ℓ                                                                    )                                                                                                  ne                                  +                                  2                                                                                            -                                                                                                (                                                                      q                                    ℓ                                                                    )                                                                                                  ne                                  +                                  1                                                                                                                      ]                                                    ⁢                                                                                                                    t                                n                                                            -                                                              t                                                                  ne                                  +                                  1                                                                                                                                                                    +                                                                                                                                                                          ∑                                                      j                            =                                                          ne                              +                              3                                                                                n                                                ⁢                                                                              [                                                                                                                            (                                                                      q                                    ℓ                                                                    )                                                                j                                                            -                                                                                                (                                                                      q                                    ℓ                                                                    )                                                                                                  j                                  -                                  1                                                                                                                      ]                                                    ⁢                                                                                                                    t                                n                                                            -                                                              t                                                                  j                                  -                                  1                                                                                                                                                                                                                                        ]                                      .                                              (        28        )            or written as:
                                          (                          Δ              ⁢                                                          ⁢                              p                res                                      )                    n                =                  141.2          ⁢                      (            2            )                    ⁢                      (            0.02878            )                    ⁢                      1                                          h                p                            ⁢                              L                f                            ⁢                              k                                              ⁢                                                                      μ                                      ϕ                    ⁢                                                                                  ⁢                                          c                      t                                                                                  ⁡                              [                                                                                                                                                                                        (                                                              q                                ℓ                                                            )                                                                                      ne                              +                              2                                                                                ⁢                                                                                                                    t                                n                                                            -                                                              t                                                                  ne                                  +                                  1                                                                                                                                                                    +                                                                                                                                                                                                      ∑                                                          j                              =                                                              ne                                +                                3                                                                                      n                                                    ⁢                                                                                    [                                                                                                                                    (                                                                          q                                      ℓ                                                                        )                                                                    j                                                                -                                                                                                      (                                                                          q                                      ℓ                                                                        )                                                                                                        j                                    -                                    1                                                                                                                              ]                                                        ⁢                                                                                                                            t                                  n                                                                -                                                                  t                                                                      j                                    -                                    1                                                                                                                                                                                                      +                                                                                                                                                                                                      (                                                          q                              ℓ                                                        )                                                                                ne                            +                            1                                                                          ⁢                                                                              t                            n                                                                          ⁢                                                  (                                                      1                            -                                                                                          1                                -                                                                                                      t                                                                          ne                                      +                                      1                                                                                                                                            t                                    n                                                                                                                                                                                )                                                                                                                    ]                                      .                                              (        28        )            
With Eq. 17 substituted for leakoff rate and Eq. 19 for the ratio of permeable to total fracture area, the reservoir pressure difference at any time tn is written as:
                                          (                          Δ              ⁢                                                          ⁢                              p                res                                      )                    n                =                                            141.2              ⁢                              (                2                )                            ⁢                              (                0.02878                )                            ⁢                              (                24                )                                      5.615                    ⁢                      1                                          r                p                            ⁢                              S                f                            ⁢                              k                                              ⁢                                                                      μ                                      ϕ                    ⁢                                                                                  ⁢                                          c                      t                                                                                  ⁡                              [                                                                                                                                                          d                                                          ne                              +                              2                                                                                ⁢                                                                                                                    t                                n                                                            -                                                              t                                                                  ne                                  +                                  1                                                                                                                                                                    +                                                                                                                                                                                                      ∑                                                          j                              =                                                              ne                                +                                3                                                                                      n                                                    ⁢                                                                                    [                                                                                                d                                  j                                                                -                                                                  d                                                                      j                                    -                                    1                                                                                                                              ]                                                        ⁢                                                                                                                            t                                  n                                                                -                                                                  t                                                                      j                                    -                                    1                                                                                                                                                                                                      +                                                                                                                                                                          d                                                      ne                            +                            1                                                                          ⁢                                                                              t                            n                                                                          ⁢                                                  (                                                      1                            -                                                                                          1                                -                                                                                                      t                                                                          ne                                      +                                      1                                                                                                                                            t                                    n                                                                                                                                                                                )                                                                                                                    ]                                      .                                              (        29        )            4) Specialized Cartesian Graph for Determining Permeability and Fracture-Face Resistance
Eq. 2 defines the total pressure difference between a point in the fracture and a point in the undisturbed reservoir as the sum of the reservoir and fracture-face pressure differences, which is written as:
                                          (                          Δ              ⁢                                                          ⁢              p                        )                    n                =                                                            141.2                ⁢                                  (                  2                  )                                ⁢                                  (                  0.02878                  )                                ⁢                                  (                  24                  )                                            5.615                        ⁢                          1                                                r                  p                                ⁢                                  S                  f                                ⁢                                  k                                                      ⁢                                                            μ                                      ϕ                    ⁢                                                                                  ⁢                                          c                      t                                                                                  ⁡                              [                                                                                                                                                          d                                                          ne                              +                              2                                                                                ⁢                                                                                                                    t                                n                                                            -                                                              t                                                                  ne                                  +                                  1                                                                                                                                                                    +                                                                                                                                                                                                      ∑                                                          j                              =                                                              ne                                +                                3                                                                                      n                                                    ⁢                                                                                    [                                                                                                d                                  j                                                                -                                                                  d                                                                      j                                    -                                    1                                                                                                                              ]                                                        ⁢                                                                                                                            t                                  n                                                                -                                                                  t                                                                      j                                    -                                    1                                                                                                                                                                                                      +                                                                                                                                                                          d                                                      ne                            +                            1                                                                          ⁢                                                                              t                            n                                                                          ⁢                                                  (                                                      1                            -                                                                                          1                                -                                                                                                      t                                                                          ne                                      +                                      1                                                                                                                                            t                                    n                                                                                                                                                                                )                                                                                                                    ]                                              +                                                    141.2                ⁢                                  (                  π                  )                                ⁢                24                            5.615                        ⁢                                          R                0                                                              r                  p                                ⁢                                  S                  f                                                      ⁢                          d              n                        ⁢                                                            t                  n                                                  t                  ne                                                                                        (        30        )            
Algebraic manipulation allows Eq. 30 to be written as:
                                                        (                              Δ                ⁢                                                                  ⁢                p                            )                        n                                              d              n                        ⁢                                          t                n                                      ⁢                                          t                ne                                                    =                                            141.2              ⁢                              (                2                )                            ⁢                              (                0.02878                )                            ⁢                              (                24                )                                      5.615                    ⁢                      1                                          r                p                            ⁢                              S                f                            ⁢                              k                                              ⁢                                    μ                              ϕ                ⁢                                                                  ⁢                                  c                  t                                                              ⁢                                                                 [                                                                                                                                                                                        d                                                              ne                                +                                2                                                                                                                    d                              n                                                                                ⁢                                                                                    (                                                                                                                                    t                                    n                                                                    -                                                                      t                                                                          ne                                      +                                      1                                                                                                                                                                                                            t                                    n                                                                    ⁢                                                                      t                                    ne                                                                                                                              )                                                                                      1                              /                              2                                                                                                      +                                                                                                                                                                                                      ∑                                                          j                              =                                                              ne                                +                                3                                                                                      n                                                    ⁢                                                                                                                    [                                                                                                      d                                    j                                                                    -                                                                      d                                                                          j                                      -                                      1                                                                                                                                      ]                                                                                            d                                n                                                                                      ⁢                                                                                          (                                                                                                                                            t                                      n                                                                        -                                                                          t                                                                              j                                        -                                        1                                                                                                                                                                                                                        t                                      n                                                                        ⁢                                                                          t                                      ne                                                                                                                                      )                                                                                            1                                /                                2                                                                                                                                    +                                                                                                                                                                                                      d                                                          ne                              +                              1                                                                                                                                          d                              n                                                        ⁢                                                                                          t                                ne                                                                                                                                    ⁢                                                  (                                                      1                            -                                                                                          1                                -                                                                                                      t                                                                          ne                                      +                                      1                                                                                                                                            t                                    n                                                                                                                                                                                )                                                                                                                    ]                            +                                                                    141.2                    ⁢                                          (                      π                      )                                        ⁢                    24                                    5.615                                ⁢                                                      R                    0                                                                              r                      p                                        ⁢                                          S                      f                                                                      ⁢                                  1                                      t                    ne                                                                                                          (        31        )            
In view of Eq. 16, the term dne+1 can be written in an alternative form as:
                                          d                          ne              +              1                                =                                                    5.615                24                            ⁢                                                S                  f                                                  A                  f                                            ⁢                              24                5.615                            ⁢                                                A                  f                                                  S                  f                                            ⁢                              d                                  ne                  +                  1                                                      =                                          5.615                24                            ⁢                                                S                  f                                                  A                  f                                            ⁢                              q                                  ne                  +                  1                                                                    ,                            (        32        )            but recognizing that qne=qne+1 and VLne=(ql)netne allows Eq. 32 to be written as:
                                          d                          ne              +              1                                =                                    5.615              24                        ⁢                                          S                f                                            t                ne                                      ⁢                                          V                Lne                                            A                f                                                    ,                            (        33        )            where VLne is the leakoff volume at the end of the injection. Define lost width due to leakoff at the end of the injection as:
                                          w            L                    ≡                                    V              Lne                                      A              f                                      ,                            (        34        )            and Eq. 33 can be written as:
                              d                      ne            +            1                          =                              5.615            24                    ⁢                      S            f                    ⁢                      w            L                    ⁢                                    1                              t                ne                                      .                                              (        35        )            
Define:
                                          c            1                    ≡                                    μ                              ϕ                ⁢                                                                  ⁢                                  c                  t                                                                    ,                            (        36        )                                                      c            2                    ≡                                    5.165              24                        ⁢                          S              f                        ⁢                          w              L                        ⁢                                          μ                                  ϕ                  ⁢                                                                          ⁢                                      c                    t                                                                                      ,                            (        37        )                                                      y            n                    ≡                                                    (                                  Δ                  ⁢                                                                          ⁢                  p                                )                            n                                                      d                n                            ⁢                                                t                  n                                            ⁢                                                t                  ne                                                                    ,                            (        38        )                                                      x            n                    ≡                      [                                                                                                                              c                        1                                            ⁡                                              [                                                                                                                                                                                                                                                                                                d                                                                                  ne                                          +                                          2                                                                                                                                                            d                                        n                                                                                                              ⁡                                                                          [                                                                                                                                                                    t                                            n                                                                                    -                                                                                      t                                                                                          ne                                              +                                              1                                                                                                                                                                                                                                                            t                                            n                                                                                    ⁢                                                                                      t                                            ne                                                                                                                                                              ]                                                                                                                                            1                                    /                                    2                                                                                                  +                                                                                                                                                                                                                                          ∑                                                                      j                                    =                                                                          ne                                      +                                      3                                                                                                        n                                                                ⁢                                                                                                                                  ⁢                                                                                                                                            [                                                                                                                        d                                          j                                                                                -                                                                                  d                                                                                      j                                            -                                            1                                                                                                                                                              ]                                                                                                              d                                      n                                                                                                        ⁢                                                                                                            (                                                                                                                                                                    t                                            n                                                                                    -                                                                                      t                                                                                          j                                              -                                              1                                                                                                                                                                                                                                                            t                                            n                                                                                    ⁢                                                                                      t                                            ne                                                                                                                                                              )                                                                                                              1                                      /                                      2                                                                                                                                                                                                                                          ]                                                              +                                                                                                                                                                  c                        2                                                                                              d                          n                                                ⁢                                                  t                          ne                                                      3                            /                            2                                                                                                                ⁢                                          (                                              1                        -                                                                              1                            -                                                                                          t                                                                  ne                                  +                                  1                                                                                                                            t                                n                                                                                                                                                        )                                                                                            ]                          ,                            (        39        )                                                      m            M                    ≡                                                    141.2                ⁢                                  (                  2                  )                                ⁢                                  (                  0.02878                  )                                ⁢                                  (                  24                  )                                            5.615                        ⁢                          1                                                r                  p                                ⁢                                  S                  f                                ⁢                                  k                                                                    ,                            (        40        )                        and                                                                b          M                ≡                                            141.2              ⁢                              (                π                )                            ⁢              24                        5.615                    ⁢                                    R              0                                                      r                p                            ⁢                              S                f                                              ⁢                                    1                              t                ne                                      .                                              (        41        )            
Combining Eq. 31 and Eqs. 36 through 41 results in:yn=mMxn+bM.  (42)
Eq. 42 suggests a graph of yn versus xn using the observed fracture-injection/falloff before-closure data will result in a straight line with the slope a function of permeability and the intercept a function of fracture-face resistance. Eqs. 41 and 42 are used to determine permeability and fracture-face resistance from the slope and intercept of a straight-line through the observed data.
5) Before-Closure Pressure-Transient Leakoff Analysis in a Dual-Porosity Reservoir System
In the present application, dual porosity refers to a mathematical model of a naturally fractured reservoir system. In paper SPE 28690, Ehlig-Economides, Fan, and Economides formulated the Mayerhofer and Economides model for dual-porosity reservoirs using Cinco-Ley and Meng's dimensionless pressure. In a paper SPE 18172: “Pressure Transient Analysis of Wells With Finite Conductivity Vertical Fractures in Dual Porosity Reservoirs,” presented at the 1988 SPE Annual Technical Conference and Exhibition, Houston, Tex., 2–5 Oct. 1988, Cinco-Ley and Meng determine dimensionless pressure with an early-time approximation for flow of a slightly compressible fluid from an infinite-conductivity fracture as:
                                          p                                          L                f                            ⁢              D                                =                                                    π                ⁢                                                                  ⁢                                  t                                                            L                      f                                        ⁢                    D                                                              ω                                      ,                            (        43        )            where for dual-porosity reservoirs,
                                          p                                          L                f                            ⁢              D                                =                                                    k                                  f                  ⁢                                                                          ⁢                  b                                            ⁢                              h                p                            ⁢              Δ              ⁢                                                          ⁢                              p                res                                                    141.2              ⁢                              q                                  L                  f                                            ⁢              B              ⁢                                                          ⁢              μ                                      ,                            (        44        )                                                      t                                          L                f                            ⁢              D                                =                      0.0002637            ⁢                                                            k                                      f                    ⁢                                                                                  ⁢                    b                                                  ⁢                t                                            ϕ                ⁢                                                                  ⁢                μ                ⁢                                                                  ⁢                                  c                  t                                ⁢                                  L                  f                  2                                                                    ,                            (        45        )            and ω is the natural fracture storativity ratio as defined by Warren, J. E. and Root, P. J. in a journal SPEJ (September 1963) on page 245: “The Behavior of Naturally Fractured Reservoirs”.
Writing Eq. 43 asωpLfD=√{square root over (πωtLfD)},  (46)and repeating the derivation for the reservoir pressure difference results in changing the final slope definition, Eq. 40, to:
                                          m            M                    ≡                                                    141.2                ⁢                                  (                  2                  )                                ⁢                                  (                  0.02878                  )                                ⁢                                  (                  24                  )                                            5.615                        ⁢                                          1                                                      r                    p                                    ⁢                                      S                    f                                    ⁢                                                            ω                      ⁢                                                                                          ⁢                                              k                                                  f                          ⁢                                                                                                          ⁢                          b                                                                                                                                .                                                            (          47          )                      ⁢        
In a dual-porosity reservoir or in a naturally fractured reservoir system, before-closure pressure-transient leakoff analysis using the specialized Cartesian graph results in an estimate of ωkfb. Methods as used in the prior art allow the product to be evaluated without an acceptable accuracy, and estimating fracture storativity ω or bulk-fracture permeability kfb requires additional testing which would involve additional inaccuracy. Therefore, since the permeability and fracture-face resistance evaluations cannot be directly obtained and since the additional testing increase the error of these evaluations, it is necessary to determine the product ωkfb with more accuracy.
Henceforth, there is a need to find another approach that mitigates nonideal leakoff behavior attributed to pressure-dependent fluid properties with more accuracy. For example, in low pressure gas reservoirs, that is, in many gas reservoirs with a pore pressure less than about 3000 psi, reservoir fluid properties are strong functions of pressure. When fluid properties are strong functions of pressure, assuming constant properties for use in pressure and time formulations will cause significant error in permeability and fracture-face resistance determinations.
These approximations as used in the prior art are therefore unsatisfactory. Thus, there is a desire not only for estimating accurate permeability and fracture-face resistance of a reservoir to appraise its quality but also for avoiding the delays linked with this type of measurements which are often very long and incompatible with the reactivity required for the success of such appraisal developments. New, faster and accurate evaluation means are therefore sought as a decision-making support.